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<p>This paper compiles and expands upon the author’s and his co-authors’ explicit calculations of the mod <i>p</i> Morava K-theory for various finite <i>p</i>-groups, a body of work currently scattered across different publications. The primary focus is on the author’s observations regarding the properties of formal group laws and the transfer in Morava K-theory. Using specific examples, this work aims to clarify the complex issues surrounding the multiplicative structure and the representation of Gröbner bases in terms of Chern classes and their transfers. A key computational question remains: is the mod 2 Morava K-theory of any finite 2-group completely generated by Chern classes and their transfers? While this conjecture by Hopkins et al. (J Am Math Soc 13:553–594, 2000), inspired by their generalized character theory of finite <i>p</i>-groups, was disproven for the mod <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_384_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(p&gt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&gt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> case by a counterexample in Kriz (Topology 36:1247–1273, 1997), the mod 2 case remains an open problem.</p>

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Morava K-theory rings for finite groups

  • 期刊论文

  • 发表日期：

    2025年10月16日

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  • Malkhaz Bakuradze

摘要

This paper compiles and expands upon the author’s and his co-authors’ explicit calculations of the mod p Morava K-theory for various finite p-groups, a body of work currently scattered across different publications. The primary focus is on the author’s observations regarding the properties of formal group laws and the transfer in Morava K-theory. Using specific examples, this work aims to clarify the complex issues surrounding the multiplicative structure and the representation of Gröbner bases in terms of Chern classes and their transfers. A key computational question remains: is the mod 2 Morava K-theory of any finite 2-group completely generated by Chern classes and their transfers? While this conjecture by Hopkins et al. (J Am Math Soc 13:553–594, 2000), inspired by their generalized character theory of finite p-groups, was disproven for the mod \(p>2\) p > 2 case by a counterexample in Kriz (Topology 36:1247–1273, 1997), the mod 2 case remains an open problem.